Task Sets: Prove angle sum formulas - Common Core Math - HS Functions

A

Warm-Up: Review What You Know

These problems review skills from earlier lessons.

1.

What is the exact value of $\sin(30\degree)$ ?

A.

$\frac{1}{2}$

B.

$\frac{\sqrt{2}}{2}$

C.

$\frac{\sqrt{3}}{2}$

D.

$1$

2.

What is the exact value of $\cos(45\degree)$ ?

A.

$\frac{1}{2}$

B.

$\frac{\sqrt{2}}{2}$

C.

$\frac{\sqrt{3}}{2}$

D.

$\frac{1}{\sqrt{3}}$

3.

The Pythagorean identity says that for any angle $\theta$ :

A.

$\sin(\theta) + \cos(\theta) = 1$

B.

$\sin^2(\theta) + \cos^2(\theta) = 1$

C.

$\sin^2(\theta) - \cos^2(\theta) = 1$

D.

$\sin(\theta) \cdot \cos(\theta) = 1$

B

Fluency Practice

C

Mixed Practice

1.

A student wants to compute $\cos(A - B)$ . Which expression is correct?

A.

$\cos(A)\cos(B) + \sin(A)\sin(B)$

B.

$\cos(A)\cos(B) - \sin(A)\sin(B)$

C.

$\cos(A) - \cos(B)$

D.

$\sin(A)\cos(B) - \cos(A)\sin(B)$

2.

The proof of the cosine subtraction formula places two angles on the unit circle. Point $P = (\cos(A), \sin(A))$ and point $Q = (\cos(B), \sin(B))$ . The squared distance $PQ^2$ expands using the distance formula: $PQ^2 = (\cos A - \cos B)^2 + (\sin A - \sin B)^2$ . Expanding the squares and using the Pythagorean identity gives $PQ^2 = \underline{\hspace{5em}} - 2\cos(A)\cos(B) - 2\sin(A)\sin(B)$ . Since the same chord length occurs when angle $A - B$ is in standard position from $(1, 0)$ to $(\cos(A - B), \sin(A - B))$ , we get $PQ^2 = \underline{\hspace{5em}} - 2\cos(A - B)$ . Setting these equal and simplifying yields $\cos(A - B) = \cos(A)\cos(B) + \underline{\hspace{5em}}\sin(A)\sin(B)$ .

constant after Pythagorean simplification (first form):

constant after Pythagorean simplification (second form):

sign in front of sin(A)sin(B):

3.

Find the exact value of $\sin(15\degree)$ . Express your answer as a single simplified fraction with a radical numerator.

4.

Compute $\tan(75\degree)$ using the formula. Decompose $75\degree = 45\degree + 30\degree$ . Recall $\tan(45\degree) = 1$ and $\tan(30\degree) = \frac{1}{\sqrt{3}}$ . Apply the formula $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$ . The numerator becomes $1 + \frac{1}{\sqrt{3}}$ , and the denominator becomes $1 - \underline{\hspace{5em}}$ . After multiplying numerator and denominator by $\sqrt{3}$ , the simplified expression is $\frac{\sqrt{3} + 1}{\sqrt{3} - \underline{\hspace{5em}}}$ .

product tan(45) tan(30):

constant in denominator after multiplying by sqrt(3):

5.

Which expression equals $\sin(2A)$ , the double angle formula for sine?

A.

$2\sin(A)\cos(A)$

B.

$\sin^2(A) + \cos^2(A)$

C.

$2\sin(A)$

D.

$\sin(A)\cos(A)$

D

Word Problems

1.

A surveyor needs the exact value of $\cos(75\degree)$ for a calculation, not a decimal approximation. She decides to express $75\degree$ as a sum of standard angles where exact values are known.

Find the exact value of $\cos(75\degree)$ using the decomposition $75\degree = 45\degree + 30\degree$ . Express your answer as a single simplified fraction with a radical numerator.

2.

To plan a triangulation measurement, a technician needs exact values of trigonometric functions at $105\degree$ and $165\degree$ . Both angles can be expressed as sums of standard angles.

1.

Find the exact value of $\sin(105\degree)$ using $105\degree = 60\degree + 45\degree$ . Express your answer as a single simplified fraction with a radical numerator.

2.

Find the exact value of $\cos(165\degree)$ using $165\degree = 120\degree + 45\degree$ . Recall $\cos(120\degree) = -\frac{1}{2}$ and $\sin(120\degree) = \frac{\sqrt{3}}{2}$ . Express your answer as a single simplified fraction with a radical numerator.

3.

For an angle $A$ in the first quadrant, $\sin(A) = \frac{3}{5}$ . A technician needs $\sin(2A)$ for a follow-up calculation.

Find $\sin(2A)$ as an exact fraction.

4.

A physics student needs $\tan(15\degree)$ as an exact value. She writes $15\degree = 45\degree - 30\degree$ and plans to use the tangent subtraction formula $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$ .

Find the exact value of $\tan(15\degree)$ . Express your answer in the simplified form $a - \sqrt{b}$ where $a$ and $b$ are integers.

5.

For an angle $A$ in the first quadrant, $\cos(A) = \frac{4}{5}$ .

Find $\cos(2A)$ as an exact fraction. Use the form $\cos(2A) = \cos^2(A) - \sin^2(A)$ .

E

Error Analysis

1.

Devon was asked to compute $\sin(30\degree + 60\degree)$ and wrote:

$\sin(30\degree + 60\degree) = \sin(30\degree) + \sin(60\degree) = \frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{1 + \sqrt{3}}{2}$

What is the main error in Devon's reasoning?

A.

Devon distributed sine over addition, which is not a valid step. The correct method uses the sine addition formula $\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)$ .

B.

Devon used the wrong exact value for $\sin(60\degree)$ . The correct value is $\frac{1}{2}$ , not $\frac{\sqrt{3}}{2}$ .

C.

Devon should have used the cosine addition formula instead of any sine formula.

D.

Devon's setup is valid but the arithmetic is wrong; the correct answer is $\frac{\sqrt{3} - 1}{2}$ .

2.

Asked to compute $\sin(2A)$ when $\sin(A) = \frac{1}{2}$ and $A$ is in QI, Mira wrote:

$\sin(2A) = 2 \cdot \sin(A) = 2 \cdot \frac{1}{2} = 1$

She justified this by saying that the "2" in $\sin(2A)$ is a multiplier just like in $2A$ inside the sine.

What is the main error in Mira's reasoning?

A.

Mira treated the double angle formula as $\sin(2A) = 2\sin(A)$ . The correct formula is $\sin(2A) = 2\sin(A)\cos(A)$ , derived from the addition formula by setting $B = A$ .

B.

Mira used the wrong value for $\sin(A)$ ; if $A$ is in QI then $\sin(A)$ should be $\frac{\sqrt{3}}{2}$ .

C.

Mira should have used the formula $\sin(2A) = \sin^2(A) + \cos^2(A) = 1$ .

D.

Mira's answer is correct because $\sin(2A)$ always equals 1 when $\sin(A) = \frac{1}{2}$ .

F

Challenge / Extension

1.

Show how to derive the formula for $\sin(A + B)$ starting from the cosine subtraction formula $\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B)$ and the co-function identity $\sin(\theta) = \cos\!\left(\frac{\pi}{2} - \theta\right)$ . Show every substitution step and explain in one sentence why each step is valid.

2.

For an angle $A$ in the first quadrant, $\sin(A) = \frac{5}{13}$ .